Electron backscattering from dynamical impurities in a Luttinger liquid

Electron backscattering in a Luttinger liquid with an impurity is investigated in the presence of zero point motion of the phonon lattice. The impurity can mean either a mass defect, an elastic defect, or a pinning defect. The phonon spectrum is then affected by the presence of the defect, which enters in the renormalization group (RG) equations for backscattering. The RG equation becomes dependent on the energy cutoff for finite Debye frequency ${\omega }_D$ giving rise to finite energy effects. We compute the local density of states and show how the renormalization group flow is affected by a finite ${\omega }_D$.

Figure 1

(a) An impurity atom is inserted in the atomic chain, resulting in a mass defect, (b) a structural deformation of the lattice results in an elastic defect, (c) interaction of one atom with an underlying substrate results in a pinning defect.

Figure 2

Backscattering flow diagram for a mass defect and two different values of the cutoff, $\Lambda ={\omega }_D$ (left) and $\Lambda =0.2{\omega }_D$ (right). The lower gray zones are unphysical, since they correspond to a negative substitute mass. Unlike in the elastic defect case, here the defect resonance always remains in the unphysical region.

Figure 3

Flow diagrams from a given initial backscattering term as function of energy. Top: defect which changes locally the mass with $\Delta m=1$. Bottom: defect which pins locally the lattice, with $\Delta K=0.1$. Different curves correspond to different $K_{\rho }$ electron interaction parameters. Velocities were chosen to be $u=2c$ and $b$ was set to a constant 2% below the Wentzel-Bardeen instability.

Figure 4

(Color online) Backscattering flow diagram for an elastic defect and three different values of the cutoff, $\Lambda ={\omega }_D$ (left), $\Lambda =0.2{\omega }_D$ (right) and the case of very small cutoff $\Lambda =0.01{\omega }_D$ (lower plot). The white and black regions denote relevant and irrelevant flow of the backscattering respectively. All gray regions mean either an unstable system [above the dashed line, see discussion after Eq. (16)] or an unphysical situation ($\Delta K<-K$, below the dashed line). The two tones of gray in those regions have only a mathematical relevant/irrelevant meaning. The local instability boundary (bounded by the dashed line) corresponds to $-1<\Delta K∕K<\Delta K_{\mathrm{crit}}∕K=-{(v_{+}{v}_{-}∕uc)}^2$. In the lower plot we have emphasized the irrelevant backscattering sliver responsible for the effective suppression at finite energies of elastic defect backscattering close to $\Delta K_{\mathrm{crit}}$.

Figure 5

Backscattering flow diagram for a defect which elastically pins the lattice at a given site (due, e.g., to the interaction with a substrate).

Figure 6

Flow diagrams from a given initial backscattering term as function of energy. The defect changes locally the elastic constant. In the inset $\Delta K>0$ while in the main plot $\Delta K$ was set to a negative value closer than 1% to the critical $\Delta K$ which is $\Delta K=-0.4375$ for the parameters used. Velocities are $u=2c$ and $b=1.5$ is a constant for all curves.